Set of all perfect squares

Question

I have been going through Velleman's How to prove book and they have explained the set of all perfect squares using this set:

$S = \{ n^2 | n \in N\}$

Then it is claimed that the set S can also be rewritten as:

$S =\{x | \exists n \in N(x = n^2)\}$

I'm not able to understand the second form of the set S. In the first set, the elementhood test was $n \in N$. So any number n belonging to N will be in set S. But in the second form, the elementhood test is $\exists n \in N(x = n^2)$, but here there is no way of knowing about the result of the equality test $x = n^2$ because we don't have about the value of x yet. So how are both the set similar ?

Answer 1

The first definition says this: $S$ is the set of all squares of natural numbers.

The second definition says this: $S$ is the set of all $x$ such that $x$ is the square of some natural number.